recursive definition odd intergers

Question

the question is "give a recursive definition for the set of all odd integers"

I know that in if I have the set $\{\cdots, -7,-5,-3,-1,1,3,5,7,9, \cdots\}$ that you have to do $x+2$ to get the positive odd integers and $x-2$ to get the negative odd integers.

What I don't know how to do is show that

Would my base case be "$1$ is in set $O$" ($O$ stands for odd integers)

What would the inductive case be?

Answer 1

Depends on how rigorous you need it. One possibility would be:

1. 1 is in $O$
2. If $n\in O$ then $n+2\in O$ and $n-2\in O$
3. $O$ is the intersection of all the sets satisfying 1) and 2)